Socolar–Taylor tile

The Socolar–Taylor tile is a single tile which is aperiodic on the Euclidean plane, meaning that it admits only non-periodic tilings of the plane, with rotations and reflections of the tile allowed.[1] It is the first known example of a single aperiodic tile, or "einstein".[2] The basic version of the tile is a simple hexagon, with printed designs to enforce a local matching rule, regarding how the tiles may be placed.[3] This rule cannot be geometrically implemented in two dimensions while keeping the tile a connected set.[2][3]

This is, however, possible in three dimensions, and in their original paper Socolar and Taylor suggest a three-dimensional analogue to the monotile.[1] Taylor and Socolar remark that the 3D monotile aperiodically tiles three-dimensional space. However the tile does allow tilings with a period, shifting one (non-periodic) two dimensional layer to the next, and so the tile is only ″weakly aperiodic″. Physical copies of the three-dimensional tile could not be fitted together without allowing reflections, which would require access to four-dimensional space.[4] [2]

Gallery

The monotile implemented geometrically. Black lines are included to show how the structure is enforced. 
A three-dimensional analogue of the Socolar-Taylor tile, without decoration (all matching rules implemented geometrically). 
A three-dimensional analogue of the monotile, with matching rules implemented geometrically. Red lines are included only to illuminate the structure of the tiling. Note that this shape remains a connected set. 
A partial tiling of three-dimensional space with the 3D monotile. 
A tiling of 3D space with one tile removed to demonstrate the structure. 

References

  1. 1 2 Socolar, Joshua E. S.; Taylor, Joan M. (2011), "An aperiodic hexagonal tile", Journal of Combinatorial Theory, Series A, 118 (8): 2207–2231, arXiv:1003.4279Freely accessible, doi:10.1016/j.jcta.2011.05.001, MR 2834173.
  2. 1 2 3 Socolar, Joshua E. S.; Taylor, Joan M. (2012), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer, 34 (1): 18–28, arXiv:1009.1419Freely accessible, doi:10.1007/s00283-011-9255-y, MR 2902144
  3. 1 2 Frettlöh, Dirk. "Hexagonal aperiodic monotile". Tilings Encyclopedia. Retrieved 3 June 2013.
  4. ↑ Harriss, Edmund. "Socolar and Taylor's Aperiodic Tile". Maxwell's Demon. Retrieved 3 June 2013.

External links

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